I can see that clearly if you take $\mathbb{Z}[x]/7\mathbb{Z}[x]$ and then quotient it by $(x^2 + 1)$ all the remaining elements will be of form $ax + b$ where $a$ can be 7 things and $b$ can be 7 things, so this "object" has 49 elements. I'm not sure how to go about showing that $\mathbb{Z}[x]/(7, x^2 + 1)$ is isomorphic to this "object" and furthermore, that this object is necessarily a field. Any help you could give would be much appreciated. Thanks in advance.